Appendix D
The Gibbs-Duhem Equation

A brief discussion of the Gibbs-Duhem equation is given in Sec. 2.4 and some applications are given in Chap. 6. In this appendix we give a derivation of the fundamental equation and we present special forms of the equation as applied to activity coefficients.

Let M be an extensive property of a mixture. For a homogeneous phase, M is a function of temperature, pressure, and mole numbers. The total differential of M is given by

(D-1)

where

(D-2)

The extensive property M is related to the partial molar properties ₁, ₂,…, by Euler’s theorem:

(D-3)

Differentiation of Eq. (D-3) gives

(D-4)

Equations (D-1) and (D-4) yield the general Gibbs-Duhem equation:¹

¹ Notice that the general Gibbs-Duhem equation applies to any extensive property, not just to the Gibbs energy, as discussed in Sec. 2.4.

(D-5)

Suppose M is the Gibbs energy G. As indicated in Chap. 2,

(D-6)

(D-7)

(D-8)

where μ, is the chemical potential. Equation (D-5) then becomes

(D-9)

In terms of excess functions for one mole of mixture, Eq. (D-9) is

(D-10)

The excess chemical potential of component i is related to activity coefficient γ_i by

(D-11)

At constant temperature and pressure, we then have

(D-12)

The phase rule tells us that in a binary, two-phase system it is not possible to change the composition of either phase while holding both temperature and pressure constant. Therefore, in a binary system, experimental data used to compute activity coefficients may be either isothermal or isobaric but not both. Therefore, Eq. (D-12) is not strictly applicable to activity coefficients for a binary system. To obtain an equation similar to, but less restrictive than Eq. (D-12), we consider how Eq. (D-9) can be rewritten in terms of activity coefficients. First, we treat the case of constant pressure and variable temperature and second, the case of constant temperature and variable pressure.

The Isobaric, Nonisothermal Case. Equation (D-9), on a molar basis, now is

(D-13)

where s is the entropy per mole of mixture. Introducing the activity coefficient,

(D-14)

where superscript 0 stands for the standard state where

a_i = γ_ix_i = 1

Differentiating Eq. (D-14) and rearranging, we have

(D-15)

Next, we utilize the Gibbs-Helmholtz equation,

(D-16)

and the mathematical identity

(D-17)

Substituting Eqs. (D-16) and (D-17) into Eq. (D-15) and solving for dμ_i yields

(D-18)

To simplify this result we recall that

(D-19)

Substituting Eqs. (D-18) and (D-19) into Eq. (D-13) and using the relations

we finally obtain

(D-20)

where h^E = h - Σ_ix_ih_i⁰ and h is the molar enthalpy of the mixture. Equation (D-20) is the desired result. It shows that the activity coefficients of a multicomponent system at constant pressure are related to one another through a differential equation that includes the enthalpy of mixing.

In many cases the standard state for component i is taken as pure liquid i at the temperature and pressure of the system. In that case, h^E is the enthalpy change that results upon mixing the pure liquids isothermally and isobarically to form the solution. However, in some cases when one of the components is a gaseous (or solid) solute, the standard-state fugacity for the solute is often taken to be Henry’s constant evaluated at the system temperature and pressure. In that case, for the solute, h_i⁰ = h_i^∞, the partial molar enthalpy of i in an infinitely dilute solution at system temperature and pressure.

The Isothermal, Nonisobaric Case. Equation (D-9) on a molar basis now is

(D-21)

where υ is the molar volume of the mixture.

Again, we introduce the activity coefficient

(D-22)

Differentiating Eq. (D-22) at constant temperature,

(D-23)

To say something about dμ_i⁰ we must now distinguish between two cases that we call case A and case B. These cases correspond to our choice of pressure for the standard state.

Case A. Let the standard state for component i be at the system temperature, at a fixed composition, and at some constant pressure that does not vary with composition. Then dμ_i⁰ = 0 and Eqs. (D-21) and (D-23) combine to

(D-24)

Case B. Let the standard state for component i be at the system temperature, at a fixed composition, and at the total pressure P of the system, which is not constant but varies with composition. In this case,

(D-25)

where υ_i⁰ is the molar volume of component i in its standard state.

Substitution of Eqs. (D-23) and (D-25) into Eq. (D-21) now gives

(D-26)

where

Equations (D-24) and (D-26) are the desired result. They show that activity coefficients of a multicomponent system at constant temperature are related to one another through a differential equation that contains volumetric properties of the liquid mixture.

Frequently, the standard state is chosen as the pure liquid at the temperature and pressure of the mixture. In that case, υ^E is the change in volume that results when the pure liquids are mixed at constant temperature and constant (system) pressure. An alternative application of Eq. (D-26) might be for a solution of a gas (or solid) in a liquid where the standard-state fugacity of the solute may be set equal to Henry’s constant evaluated at system temperature and total pressure. In that case, for the solute, υ_i⁰ = , the partial molar volume of i in an infinitely dilute solution at system temperature and total pressure.

At low or moderate pressures, the right-hand side of Eq. (D-26) is often so small that Eq. (D-12) provides an excellent approximation.